How to simulate a suction cup?
EDIT: Completely rewrote this answer, as I found a much better solution.


Basically you can connect two rigid bodies then keyframe their connection to decide when to turn it on or off.

Here's how:
After you set up your rigid bodies, select your cube, then your suction cup object, then connect the two using the physics tab. This will constrain the cube to the cup in a rigid body responsive manner (unlike ordinary constraints that make a mess).


Then all you need to do is to select the empty that represents the relationship between the two objects, and animate the "enabled" property on the constraint.



I guess you'll just have to figure out the rest about implementing this in BGE (which enables dynamically controlling animations).

Blendfile.